Local vertices, quadratic propagators and double-copy structure of one-loop integrands from forward limits

TL;DR

This paper extends the worldsheet forward-limit construction of one-loop amplitudes to theories with gravitons by introducing double Yang-Mills-scalar (dYMS) integrands. It develops systematic localization methods (localizing nonlocal terms via X- and BCJ-patterns) to derive local multi-point vertices and obtain quadratic propagators, enabling full dYMS, EYM, and GR loop representations and tree-level analogues. A key outcome is a consistent double-copy structure at the level of tree-level effective subcurrents and loop insertions, with coefficients independent of loop momentum, which then yield quadratic propagators after partial fractioning. The framework sets the stage for a unified BCJ-inspired construction with quadratic propagators across loop and tree amplitudes, and identifies explicit vertex patterns up to $| extsf{W}|\, ext{≤}\,3$, with scalable strategies for higher $| extsf{W}|$ in future work.

Abstract

By worldsheet approach, $n$-point one-loop integrand can be expressed as a combination of $(n+2)$-point tree-level bi-adjoint scalar (BS) amplitudes under forward limit. The integrands constructed by this approach have two closely related features that are different from conventional Feynman diagrams. First, the denominators of loop propagators are linear functions of the loop momentum. Second, the local vertex expression is not manifest. In our previous work, a systematic approach was proposed to handle the nonlocal terms in the one-loop integrand of Yang-Mills-scalar (YMS) theory. Upon canceling the nonlocalities, quadratic propagator forms of both YMS and Yang-Mills (YM) integrands are naturally obtained. In this paper, we generalize the calculation to theories involving gravitons by introducing the one-loop double Yang-Mills-scalar (dYMS) integrands. The cancellation of the nonlocalities of the dYMS integrand in the forward limit coincides with the emergence of local multi-point vertices. We provide two equivalent methods for extracting the vertices and give the final expression of the dYMS integrand with quadratic propagators. In this formula, tree-level effective subcurrents, which exhibit double-copy structures, are attached to the loop propagator line via the local vertices constructed. The quadratic propagator formulas for Einstein-Yang-Mills (EYM) and gravity (GR) integrands are further derived, by the help of the formula for dYMS. The extraction of local vertices in one-loop dYMS integrand also applies at tree-level, thus we have the corresponding expressions of tree-level dYMS, EYM, and GR amplitudes.

Figures (6)

• Figure 1: In the graphic rule, (a) is a typical chain that constructs a connected graph (b). Correspondingly, when the strength tensors are expanded explicitly, the lines in (c) form refined graphs such as (d) in the refined graphic rule.
• Figure 2: The first graph-based relation
• Figure 3: The second graph-based relation
• Figure 4: For the given partition $\{A_1=\{x_1,x_3,g_1\},A_2=\{x_2,g_2\},A_3=\{x_4,g_3\}\}$, graphs (a) and (c) are decomposed into connected and disconnected subgraphs, whereas graph (b) is decomposed into connected subgraphs only.
• Figure 5: A typical right graph (a) and its topology (b)